UV grid only showing on the uv/image editor not the 3d view
Judging from your screenshot, you're using Cycles Render engine and Textured View in the viewport. Cycles in this case behaves like this: It will show the image texture node on your model which is at the moment selected in the Node Editor (in the screenshot i marked it in a red circle), no matter if it is connected in the Node Tree or not. If you click another Image Texture Node in the Node Editor, the viewport will show you this one. Not seeing any texture could either mean you have no material set up at all, or an Image Texture Node is selected which does not provide any output at the moment.

The material preview mode on the other hand will give you a representation of the "real" material output, i.e. it tries to evaluate all nodes which are connected and show the result, independent of your selection.